Defining polynomial
| \( x^{12} + 8 x^{10} + 4 x^{9} + 8 x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{4} + 8 x^{3} + 6 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $12$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $30$ |
| Discriminant root field: | $\Q_{2}(\sqrt{*})$ |
| Root number: | $1$ |
| $|\Aut(K/\Q_{ 2 })|$: | $2$ |
| This field is not Galois over $\Q_{2}$. | |
Intermediate fields
| 2.3.2.1, 2.6.11.4 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{2}$ |
| Relative Eisenstein polynomial: | \( x^{12} - 8 x^{11} - 3680 x^{10} + 103564 x^{9} + 9285648 x^{8} - 671647988 x^{7} + 20782751964 x^{6} - 344788776784 x^{5} + 2741671255044 x^{4} - 9778483798600 x^{3} + 8841639826112 x^{2} + 5746495910112 x - 1934094156586 \) |
Invariants of the Galois closure
| Galois group: | 12T223 |
| Inertia group: | 12T187 |
| Unramified degree: | $2$ |
| Tame degree: | $3$ |
| Wild slopes: | [4/3, 4/3, 8/3, 8/3, 3, 3, 19/6, 19/6] |
| Galois mean slope: | $1183/384$ |
| Galois splitting model: | $x^{12} - 10 x^{10} + 29 x^{8} - 84 x^{7} + 128 x^{6} - 124 x^{5} + 91 x^{4} - 44 x^{3} + 18 x^{2} - 4 x + 1$ |